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<meta name="twitter:description" content="暴力测试最简单的算法，从2枚举到sqrt(n)就可以知道是不是素数了。 费马小定理对于质数n和任意整数a，有a^n ≡ a(mod n)(同余)。反之，若满足a^n≡a(mod n)，n也有很大概率为质数，称n是一个基于a的伪素数。将两边同时约去一个n，则有a^(n−1)≡1(mod n)。 这个定理是来判断一个数字是不是合数的，而不是素数。如果不符费马小定理，一定是合数， 如果符合费马小定理不一">



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          <h2 class="post-title" itemprop="name headline">找出100万以内的质数</h2>
        

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        <h2 id="暴力测试"><a href="#暴力测试" class="headerlink" title="暴力测试"></a>暴力测试</h2><p>最简单的算法，从2枚举到<code>sqrt(n)</code>就可以知道是不是素数了。</p>
<h2 id="费马小定理"><a href="#费马小定理" class="headerlink" title="费马小定理"></a>费马小定理</h2><p>对于质数n和任意整数a，有<code>a^n ≡ a(mod n)</code>(同余)。<br>反之，若满足<code>a^n≡a(mod n)</code>，<code>n</code>也有很大概率为质数，称<code>n</code>是一个基于<code>a</code>的伪素数。将两边同时约去一个<code>n</code>，则有<code>a^(n−1)≡1(mod n)</code>。</p>
<p>这个定理是来判断一个数字是不是合数的，而不是素数。如果不符费马小定理，一定是合数， 如果符合费马小定理不一定是素数（但是是素数的可能性比较高）。</p>
<p>如果判定<code>n</code>为合数，那么结果一定正确。如果判定<code>n</code>为质数，那么只有当<code>n</code>是基于2的伪素数时才会出错</p>
<a id="more"></a>
<h2 id="Miller-Rabin素数检验-米勒-拉宾素数检验"><a href="#Miller-Rabin素数检验-米勒-拉宾素数检验" class="headerlink" title="Miller-Rabin素数检验(米勒-拉宾素数检验)"></a>Miller-Rabin素数检验(米勒-拉宾素数检验)</h2><p>在费马小定理基础上增加了二次判定：<br>如果<code>n</code>是奇素数，则<code>x^2 ≡ 1(mod n)</code>的解为<code>x≡1</code>或<code>x ≡ n−1(mod n)</code></p>
<p>这个x是：<code>a^(p−1)=a^(2u∗r)</code>，所以<code>p−1=2u∗r</code>，那么x自然也知道了：这个x是一系列的数字，让在指数<code>2^u</code>里拿出一个2就编程<code>2^(u-1) * 2</code>，这样不就有平方了么，剩下的底数就是x了，并且在还可以继续不停的这样操作，到<code>u = 0</code>为止。</p>
<p>Miller-Rabin素数检验仍会出错：如果是素数，不出错，如果为合数，也会小概率出错<code>2^(-s)</code>，所以多次检测，会让错误率变低，这里<code>s</code>是检测次数。</p>
<h2 id="举例说明"><a href="#举例说明" class="headerlink" title="举例说明"></a>举例说明</h2><blockquote>
<p>举个Matrix67 Blog上的例子，假设n=341，我们选取的a=2。则第一次测试时，2^340 mod 341=1。由于340是偶数，因此我们检查2^170，得到2^170 mod 341=1，满足二次探测定理。同时由于170还是偶数，因此我们进一步检查2^85 mod 341=32。此时不满足二次探测定理，因此可以判定341不为质数。</p>
</blockquote>
<h2 id="代码实现"><a href="#代码实现" class="headerlink" title="代码实现"></a>代码实现</h2><p>参考<a href="http://blog.csdn.net/alps1992/article/details/51588971" target="_blank" rel="noopener">算法基础 - 素数判定(Miller-Rabin算法)</a><br><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br></pre></td><td class="code"><pre><span class="line">伪代码：</span><br><span class="line">Miller-Rabin(n):</span><br><span class="line">    If (n &lt;= 2) Then</span><br><span class="line">        If (n == 2) Then</span><br><span class="line">            Return True</span><br><span class="line">        End If</span><br><span class="line">        Return False</span><br><span class="line">    End If</span><br><span class="line"></span><br><span class="line">    If (n mod 2 == 0) Then</span><br><span class="line">        // n为非2的偶数，直接返回合数</span><br><span class="line">        Return False</span><br><span class="line">    End If</span><br><span class="line"></span><br><span class="line">    // 我们先找到的最小的a^u，再逐步扩大到a^(n-1)</span><br><span class="line"></span><br><span class="line">    u = n - 1; // u表示指数</span><br><span class="line">    while (u % 2 == 0) </span><br><span class="line">        u = u / 2</span><br><span class="line">    End While // 提取因子2</span><br><span class="line"></span><br><span class="line">    For i = 1 .. S // S为设定的测试次数</span><br><span class="line">        a = rand_Number(2, n - 1) // 随机获取一个2~n-1的数a</span><br><span class="line">        x = a^u % n</span><br><span class="line">        tu = u</span><br><span class="line">        While (tu &lt; n) </span><br><span class="line">            // 依次次检查每一个相邻的 a^u, a^2u, a^4u, ... a^(2^k*u)是否满足二次探测定理</span><br><span class="line">            y = x^2 % n </span><br><span class="line">            If (y == 1 and x != 1 and x != n - 1)   // 二次探测定理</span><br><span class="line">                // 若y = x^2 ≡ 1(mod n)</span><br><span class="line">                // 但是 x != 1 且 x != n-1</span><br><span class="line">                Return False</span><br><span class="line">            End If</span><br><span class="line">            x = y</span><br><span class="line">            tu = tu * 2 </span><br><span class="line">        End While</span><br><span class="line">        If (x != 1) Then    // Fermat测试</span><br><span class="line">            Return False</span><br><span class="line">        End If</span><br><span class="line">    End For</span><br><span class="line">    Return True</span><br></pre></td></tr></table></figure></p>
<p>Java代码编写：<br><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> java.math.*;</span><br><span class="line"><span class="keyword">import</span> java.util.*;</span><br><span class="line"></span><br><span class="line"><span class="comment">/**</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@author</span> zxlg</span></span><br><span class="line"><span class="comment"> *</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">All_primes_in_million</span> </span>&#123;</span><br><span class="line"></span><br><span class="line">	<span class="comment">/**</span></span><br><span class="line"><span class="comment">	 * <span class="doctag">@param</span> args</span></span><br><span class="line"><span class="comment">	 */</span></span><br><span class="line">	<span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span> </span>&#123;</span><br><span class="line">		<span class="comment">// TODO Auto-generated method stub</span></span><br><span class="line">		print_all_prime_in_million(<span class="number">10000</span>);</span><br><span class="line"></span><br><span class="line">	&#125;</span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">print_all_prime_in_million</span><span class="params">(<span class="keyword">int</span> n)</span> </span>&#123;</span><br><span class="line">		ArrayList&lt;Integer&gt; arr = <span class="keyword">new</span> ArrayList&lt;Integer&gt;();</span><br><span class="line">		<span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; n; i++) &#123;</span><br><span class="line">			<span class="keyword">if</span> (Miller_Rabin(i)) &#123;</span><br><span class="line">				arr.add(i);</span><br><span class="line">			&#125;</span><br><span class="line">		&#125;</span><br><span class="line">		System.out.println(arr.toString());</span><br><span class="line">		System.out.println(n + <span class="string">"以内共有"</span> + arr.size() + <span class="string">"个素数"</span>);</span><br><span class="line">	&#125;</span><br><span class="line"></span><br><span class="line">	<span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">boolean</span> <span class="title">Miller_Rabin</span><span class="params">(<span class="keyword">int</span> n)</span> </span>&#123;</span><br><span class="line">		<span class="comment">// 小于2的情况</span></span><br><span class="line">		<span class="keyword">if</span> (n &lt;= <span class="number">2</span>) &#123;</span><br><span class="line">			<span class="keyword">if</span> (n == <span class="number">2</span>) &#123;</span><br><span class="line">				<span class="keyword">return</span> <span class="keyword">true</span>;</span><br><span class="line">			&#125;</span><br><span class="line">			<span class="keyword">return</span> <span class="keyword">false</span>;</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="comment">// 偶数</span></span><br><span class="line">		<span class="keyword">if</span> (n % <span class="number">2</span> == <span class="number">0</span>) &#123;</span><br><span class="line">			<span class="keyword">return</span> <span class="keyword">false</span>;</span><br><span class="line">		&#125;</span><br><span class="line"></span><br><span class="line">		<span class="comment">// 先找到的最小的a^u，再逐步扩大到a^(n-1)</span></span><br><span class="line"></span><br><span class="line">		<span class="keyword">int</span> u = n - <span class="number">1</span>;<span class="comment">// u表示指数</span></span><br><span class="line">		<span class="keyword">int</span> S = <span class="number">20</span>;<span class="comment">// 检测次数</span></span><br><span class="line"></span><br><span class="line">		<span class="comment">// 提取公因子2</span></span><br><span class="line">		<span class="keyword">while</span> (u % <span class="number">2</span> == <span class="number">0</span>) &#123;</span><br><span class="line">			u /= <span class="number">2</span>;</span><br><span class="line">		&#125;</span><br><span class="line"></span><br><span class="line">		<span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; S; i++) &#123;</span><br><span class="line">			<span class="comment">// 随机选取2~n-1之间的数a</span></span><br><span class="line">			<span class="keyword">int</span> a = (<span class="keyword">int</span>) (Math.random() * <span class="number">2</span> + (n - <span class="number">3</span>));</span><br><span class="line">			<span class="comment">// x = a^u % n</span></span><br><span class="line">			<span class="keyword">int</span> x = quick_power(a, u).mod(BigInteger.valueOf(n)).intValue();</span><br><span class="line">			<span class="keyword">int</span> tu = u;</span><br><span class="line">			<span class="comment">// 依次次检查每一个相邻的 a^u, a^2u, a^4u, ... a^(2^k*u)是否满足二次探测定理</span></span><br><span class="line">			<span class="keyword">while</span> (tu &lt; n) &#123;</span><br><span class="line">				<span class="keyword">int</span> y = x * x % n;</span><br><span class="line">				<span class="keyword">if</span> (y == <span class="number">1</span> &amp;&amp; x != <span class="number">1</span> &amp;&amp; x != n - <span class="number">1</span>) &#123;<span class="comment">// 二次探测定理</span></span><br><span class="line">					<span class="comment">// 若y = x^2 ≡ 1(mod n)</span></span><br><span class="line">					<span class="comment">// 但是 x != 1 且 x != n-1</span></span><br><span class="line">					<span class="keyword">return</span> <span class="keyword">false</span>;</span><br><span class="line">				&#125;</span><br><span class="line">				x = y;</span><br><span class="line">				tu = tu * <span class="number">2</span>;</span><br><span class="line">			&#125;</span><br><span class="line"></span><br><span class="line">			<span class="keyword">if</span> (x != <span class="number">1</span>) &#123; <span class="comment">// 费马测试</span></span><br><span class="line">				<span class="keyword">return</span> <span class="keyword">false</span>;</span><br><span class="line">			&#125;</span><br><span class="line">		&#125;</span><br><span class="line"></span><br><span class="line">		<span class="keyword">return</span> <span class="keyword">true</span>;</span><br><span class="line">	&#125;</span><br><span class="line"></span><br><span class="line">	<span class="comment">// 快速幂</span></span><br><span class="line">	<span class="function"><span class="keyword">public</span> <span class="keyword">static</span> BigInteger <span class="title">quick_power</span><span class="params">(<span class="keyword">int</span> num, <span class="keyword">int</span> power)</span> </span>&#123;</span><br><span class="line">		BigInteger ans = BigInteger.valueOf(<span class="number">1</span>);</span><br><span class="line">		BigInteger base = BigInteger.valueOf(num);</span><br><span class="line">		<span class="keyword">while</span> (power != <span class="number">0</span>) &#123;</span><br><span class="line">			<span class="comment">// power转为2进制，若该位为1</span></span><br><span class="line">			<span class="keyword">if</span> ((power &amp; <span class="number">1</span>) == <span class="number">1</span>) &#123;</span><br><span class="line">				ans = ans.multiply(base);</span><br><span class="line">			&#125;</span><br><span class="line">			<span class="comment">// 转换为基础值，没轮循环为原值的平方</span></span><br><span class="line">			base = base.multiply(base);</span><br><span class="line">			<span class="comment">// power取下一位</span></span><br><span class="line">			power = power &gt;&gt; <span class="number">1</span>;</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="keyword">return</span> ans;</span><br><span class="line">	&#125;</span><br><span class="line"></span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></p>
<h2 id="参考文献"><a href="#参考文献" class="headerlink" title="参考文献"></a>参考文献</h2><ol>
<li><a href="http://blog.csdn.net/alps1992/article/details/51588971" target="_blank" rel="noopener">算法基础 - 素数判定(Miller-Rabin算法)</a></li>
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    <a href="http://happylg.cn/2017/08/30/prime-numbers/" title="找出100万以内的质数">http://happylg.cn/2017/08/30/prime-numbers/</a>
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              <div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-2"><a class="nav-link" href="#暴力测试"><span class="nav-number">1.</span> <span class="nav-text">暴力测试</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#费马小定理"><span class="nav-number">2.</span> <span class="nav-text">费马小定理</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Miller-Rabin素数检验-米勒-拉宾素数检验"><span class="nav-number">3.</span> <span class="nav-text">Miller-Rabin素数检验(米勒-拉宾素数检验)</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#举例说明"><span class="nav-number">4.</span> <span class="nav-text">举例说明</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#代码实现"><span class="nav-number">5.</span> <span class="nav-text">代码实现</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#参考文献"><span class="nav-number">6.</span> <span class="nav-text">参考文献</span></a></li></ol></div>
            

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          proceedsearch();
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    $('.popup-btn-close').click(onPopupClose);
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    function addCount(Counter) {
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            newcounter.save(null, {
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    });
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